Faraday's law -- circular loop with a triangle

[TSny]

That's right.

Yes.It should read the line integral along the staright line joining a and b. As per my calculations, it is (r²/2)dB/dt= ε/2π . Is that correct?

Anyway, it is NOT zero although the resistance of the ring is uniform and the voltmeter probes are in a perpendicular plane.

Yes. ε/(2π). Good.

 

[TSny]

Meanwhile it'll be bugging me too ... I wonder about the "loop" in fig. 2. Don't we get a finite amount of flux thru this loop area? The loop surface is some kind of weirdo 3-D thing isn't it?

Yes, if you close the voltmeter loop by going from b to a along the ring. And then
B dot A ≠ 0?

 

[TSny]

@TSny, suppose the resistance of this ring is uniform and the total emf induced in the ring is 10V, anticlockwise. Let's call the top and bottom terminals of the voltmeter V₂ and V₁ respectively.
View attachment 213406
So the voltmeter should read 5V, with V₂ -ve (top) and V₁ +ve (bottom).

Now, the "electrostatic voltage" between A and V₂ should not be zero even though the resistance of the wire connecting them is zero. Ditto for B and V₁.

Assuming the voltmeter doesn't draw any current, the "electrostatic potential" along the wire from B to A through the meter should look something like this.View attachment 213405
It won't be all straight lines as the induced emf varies along the length of the wire, but this gives an idea about the electrostatic voltage gradient along the voltmeter probes.

Do you agree with this?

Yes, I agree with all of this.

 

[TSny]

So VM reading is ##\varepsilon##/4 - (dB/dt)(a²/4)(π-2)
= (dB/dt){(πa²/4 - (πa²/4)(π-2)}
= (dB/dt)(a²/2) = (dB/dt)πa²/2π = ##\varepsilon##/2π
with a = radius.

Whew!

Yes, that looks good to me.

 

[timetraveller123]

sorry for the very very late reply my o levels were going on @cnh1995

Yes, that is correct. But when rude man said the voltmeter loop does not enclose any flux and when I repeated it in bold pink letters, I had this situation in mind, where the voltmeter probes are parallel to the B-field. Here, the voltmeter will read zero if the resistance of the ring is uniform.
View attachment 213396

I was going to edit it after reading #63, but somehow, I forgot it.
Thanks!

ok in this case there is no electrostatic voltage due to uniform resistance then to calculate voltmeter reading it is
there is (somewhat of not formally speaking) rise of ## \frac{\epsilon}{2}## and a drop of ir and ir also equals ##\frac{\epsilon}{2}## thus voltmeter reads am i right?

ok then as for problem by @TSny i understand figure 1 it is emf/4 then for figure 2 looking from on top
the area enclosed like what @rude man said is ##\frac{a^2}{4}(\pi - 2)## then the equations become
as what rude man i got the same result also
are my reasonings correct?

 

[cnh1995]

sorry for the very very late reply my o levels were going on @cnh1995

ok in this case there is no electrostatic voltage due to uniform resistance then to calculate voltmeter reading it is
there is (somewhat of not formally speaking) rise of ## \frac{\epsilon}{2}## and a drop of ir and ir also equals ##\frac{\epsilon}{2}## thus voltmeter reads am i right?

ok then as for problem by @TSny i understand figure 1 it is emf/4 then for figure 2 looking from on top
the area enclosed like what @rude man said is ##\frac{a^2}{4}(\pi - 2)## then the equations become
as what rude man i got the same result also
are my reasonings correct?

Yes, if you've got the same results as we did.

 

[timetraveller123]

@chn1995 thanks now returning to walter lewin problem

say the magnetic field is causing emf e in the clockwise direction

by symetry the induced emf along ab and cd will be the same

but to make the current same in the circuit electrostatic voltage will develop across a and b and c and d to even out the current

current induced in entire circuit is ##\frac{e}{r_1 + r_2}##
assuming ##r_2 > r_1## then ##v_B > v_A## then the equation is
##
\frac{v_B - v_A}{r_2} + \frac{e}{2 r_2} = \frac{e}{r_1 + r_2}
##
where the voltmeter measures ##v_B - v_A##
is this correct?

 

[cnh1995]

@chn1995 thanks now returning to walter lewin problem

View attachment 215129
say the magnetic field is causing emf e in the clockwise direction

by symetry the induced emf along ab and cd will be the same

but to make the current same in the circuit electrostatic voltage will develop across a and b and c and d to even out the current

current induced in entire circuit is ##\frac{e}{r_1 + r_2}##
assuming ##r_2 > r_1## then ##v_B > v_A## then the equation is
##
\frac{v_B - v_A}{r_2} + \frac{e}{2 r_2} = \frac{e}{r_1 + r_2}
##
where the voltmeter measures ##v_B - v_A##
is thius correct?

The voltmeter reads either ir₁ or ir₂ depending on where it is placed. It is not necessarily V_A-V_B.

 

[timetraveller123]

oh wait sorry i got mixed with voltmeter readings and electrostatic voltage
the first term is electrostatic voltage
the second term is em induced emf
the right hand side is net emf
is that correct

 

[mabilde]

Find the right answer on:
Greetings from Belgium, Cyriel Mabilde

 

[rude man]

I need to study the video at least 1-2 more times but it reinforces the interpretation I had before engaging in this lengthy thread. Thank you so much. We owe you a great debt of gratitude.

Perhaps you care to comment on another area of disagreement I have with Dr Lewin's lecture, where he presents a battery in series with a resistor. He claims there is an E field in the battery such that the integral of the E field around the loop is zero. I disagree and claim the net E field inside the battery, other than that accounted for by any internal resistance, is zero. Do you think it plausible that there are two equal and opposite E fields in the battery, one conservative and one non-conservative, canceling each other? (again ignoring internal resistance).

EDIT: by now I'm convinced this is the case. Proof: 2 fields, E_s and E_m. Contour integral of E_s is zero. Therefore a field E_s must exist in the battery, pointing from + to -. Also there must exist a field E_m since the battery is a source of emf, pointing - to + and of equal magnitude to E_s. That's QED.

(I don't pretend to have come to this conclusion by myself. Prof. H.H. Skilling of Stanford seems to state the same thing (indirectly) in his Fundamentals of Electric Waves (eq.185, p. 75, 2nd edition, 10th printing, John Wiley & Sons). Dr Skilling presents this equation as a prelude to electromagnetism where the E integral around a loop is of course non-zero. His obvious implication is that no source of emf by itself (e.g. without any internal resistance) has a net E field such that its own line integral is non-zero.

I find it inconceivable that Dr Skilling's textbook could have survived 9 printings and 14 years of wide use without someone having pointed out the error.)

 

[rude man]

Also Lewin's statement that "Kirchhoff is Wrong" has always puzzled me. Kirchhoff never talked about E fields, he talked only about voltage drops around any circuit = 0.

 

[Charles Link]

You may also want to see this thread: https://www.physicsforums.com/threads/walter-lewin-paradox.948122/#post-6003552 Instead of putting a voltmeter between the points A and B, you can replace the voltmeter by a large resistor, and compute the product of the current flow and the resistance. ## \\ ## Perhaps there is a shortcut, but it looks like a lengthy algebraic exercise. If you compute the current in the resistor between A and B, I think you could just take that current and multiply it by the resistor value, and that's what you get for a large resistor in parallel to it since the flux (and thereby the rate of flux change) between the large voltmeter resistor and this resistor ## r_2 ## is zero, if you put them directly parallel to each other. That greatly simplifies the algebra. You don't need to include the extra voltmeter resistor. ## \\ ## In any case, you will get a different answer if you put the voltmeter resistor parallel to ## r_2 ##, (directly adjacent to it), or across the arc with resistor ## r_1 ##. In general the answer will be path dependent.

 

[timetraveller123]

hi all this thread has cleared up a lot of doubts for me however this still some doubts i would like to clarify

how exactly does current flow?
like when it flows through the conductor there is still some non neglible resistance there which needs electric field to overcome i know the electric field in conductor is only zero when it is electrostatic but i don't understand what is different here
like when pd is applied across a conductor electron rush to negate the electric field but once they reach the end of conductor they sort of leak out of the conductor and another set of electrons comes to fill the space essentially setting up a current but at every instant there is electrons there to negate the pd so why would there be non zero electric field even when current is flowing
i am sorry if the question doesn't make sense i am quite confused and not even sure of what i don't understand thanks

 

[rude man]

It may be remarked that the Mabilde setup does not measure potentials directly. Rather, by virtue of putting the meter leads inside the B flux, a secondary emf equal to the measured potential is generated, the attendant potential drop around the meter loop being what the voltmeter measures. The data is correct since the area inscribed by the coil section plus the meter leads is the same fraction of the total coil flux area as the fraction of the potential around the coil. The setup is therefore not empirical proof of the existence of a potential drop over the measured coil segment. Also, his description of the coupling between the coil and the meter circuit is misleading, as can be readily seen by rearranging the meter leads in one of his diagrams, which would alter the degree of coupling. Cf. my recent Insight tutorial on Dr. Lewin's Conundrum.

 

[rude man]

Just to wrap up the OP’S original post::

Let
d1 = length of one arc (=2πa/3),
d2 = length of one side of the triangle,
S = area A-B,

The strategy is to irradiate (apply the B field to) the triangle area A-B-C and area A-B separately, then vectorially add the various E_s and E_m (static and emf) fields by superposition. The other segments, not abutting the A-B loop, can be ignored altogether; whether or not they are irradiated does not affect the E fields in the two A-B segments.

We first note that when just the triangle area is irradiated. there is no E_s field. This means that the triangle area can also be ignored since potential differences are line-integrated E_s fields only.

So we irradiate S only:
If E_m = emf field around the A-B loop and i the corresponding current,
emf = E_m(d1 + d2) = -S dB/dt = i(r1 + r2) so we know E_m and i;

and if
E_s1 = static field along arc AB,
E_s2 = static field along the triangle side AB,

from the basic relation j = σE, j = current density, E = E_m + E_s, we get

i = (E_m - E_s1)d1/r1 or
E_s1 = (1/d1)(E_m d1 - i r1) = E_m - i r1/d1

and similarly
E_s2 = r2 i/d2 - E_m.
Then ΔV = E_s1 d1 = E_s2 d2.
The rest is geometry and substitution.

Note that this computation is not that of a voltmeter (by all we have discussed so far) measuring voltage between A and B. The voltmeter reading is in fact VB - VA = -r1 i.

EDIT: I forgot that the bottom side of the triangle has R=2r2, not r2. That unfortunately invalidates the computation. May look into it further but it looks bad.

 

[timetraveller123]

Just to wrap up the OP’S original post::
View attachment 231017
Let
d1 = length of one arc (=2πa/3),
d2 = length of one side of the triangle,
S = area A-B,

The strategy is to irradiate (apply the B field to) the triangle area A-B-C and area A-B separately, then vectorially add the various E_s and E_m (static and emf) fields by superposition. The other segments, not abutting the A-B loop, can be ignored altogether; whether or not they are irradiated does not affect the E fields in the two A-B segments.

We first note that when just the triangle area is irradiated. there is no E_s field. This means that the triangle area can also be ignored since potential differences are line-integrated E_s fields only.

So we irradiate S only:
If E_m = emf field around the A-B loop and i the corresponding current,
emf = E_m(d1 + d2) = -S dB/dt = i(r1 + r2) so we know E_m and i;

and if
E_s1 = static field along arc AB,
E_s2 = static field along the triangle side AB,

from the basic relation j = σE, j = current density, E = E_m + E_s, we get

i = (E_m - E_s1)d1/r1 or
E_s1 = (1/d1)(E_m d1 - i r1) = E_m - i r1/d1

and similarly
E_s2 = r2 i/d2 - E_m.
Then ΔV = E_s1 d1 = E_s2 d2.
The rest is geometry and substitution.

Note that this computation is not that of a voltmeter (by all we have discussed so far) measuring voltage between A and B. The voltmeter reading is in fact VB - VA = -r1 i.

thanks for reply
i guess i was highly coufused on how actually the placement of voltmeter could affect the reading and the buildup of charge in the circuit now i get it thanks to everyone involved in this very lengthy thread
sorry for the late reply i hadn't logged in for a long time

 

[rude man]

You started quite a rolling ball didn't you! I'm glad you did because it got me to revisit and re-learn old matrial myself. Lots of fun!
PS did you ever get an answer from your instructor?

 

[timetraveller123]

You started quite a rolling ball

apparently so another similar post to this also was very several pages long
https://www.physicsforums.com/threads/induced-emf-between-two-points.926298/

but so much more of this makes more sense now that i have learned a bit of vector calculus

did you ever get an answer from your instructor?

i got it from a regional physics Olympiad paper and apparently according to my current teacher no one solved it completely during that competition

thank you for you continued interest in physics forum(i find this far superior to stack exchange ;) ) many of those in this thread have helped in many of my previous threads including you @rudeman

 

[rude man]

Good for you for persevering! That put you around 80% above the others who ask for homework help, IMO.

The key thing I'd like to impart here is the importance of understanding that there are two kinds of E fields, namely emf-generated and electrostatic. It's very confusing to analyze this kind of problem without that understanding.

If you want to pursue this further you may want to read my recent Insight article:
https://www.physicsforums.com/insights/a-new-interpretation-of-dr-walter-lewins-paradox/

Keep it up!

 

[timetraveller123]

thanks for your encouragement i will take a look at the article
but i have one more doubt
what exactly happens in a battery i am given to understand it provides a non conservative emf then shouldn't it be in the other side of kirchhoff laws

 

[rude man]

thanks for your encouragement i will take a look at the article
but i have one more doubt
what exactly happens in a battery i am given to understand it provides a non conservative emf then shouldn't it be in the other side of kirchhoff laws

I answer this question in the referenced article, but there are two E fields in the battery: one points - to + and is emf-generated (I call it E_m), and the other points + to - and is electrostatic-generated, I call it E_s. The two fields are equal and oppsite inside the battery (assuming zero internal resistance) so the net E field in the battery is zero.

As I also said earlier, Kirchoff's laws really apply to voltage drops and rises, not E fields. However, the voltage law applies also to E_s fields but not to E_m fields. That's because voltage by definition is ## V_b - V_a = \int_a^b \vec E_s \cdot d \vec l. ## So for the battery plus resistor circuit, ## \oint \vec E_s \cdot d \vec l = 0 ## but ## \oint (\vec E_s + \vec E_m) \cdot d \vec l = iR, ## i = current, R = circuit resistance. The total E field = E_s + E_m, vectorially added.

 

[timetraveller123]

I answer this question in the referenced article

actually i am coming here from there

i believe the electrostatic field is due to the buildup of charge at the resistor
ok i see what you are saying now the Es field points in the opposite direction from positive terminal to negative hence giving rise to voltage rise thanks for that
one more question

why does the voltmeter measure Es not the total E field as you said in the article

 

[rude man]

actually i am coming here from there

i believe the electrostatic field is due to the buildup of charge at the resistor
ok i see what you are saying now the Es field points in the opposite direction from positive terminal to negative hence giving rise to voltage rise thanks for that
one more question

why does the voltmeter measure Es not the total E field as you said in the article

This is something you just have to accept. A voltmeter reads voltage and voltage is the line integral of E_s, not E_m.

Think of a ring, uniform resistance, in a time-changing B field. You know the emf = -∂φ/∂t = ## \mathcal E = \oint \vec E \cdot d \vec l. ## There is no E_s field in the ring so ΔV = 0 between any two points along the ring.

BUT - the trouble is that when you attach a voltmeter by its leads to any two points along the ring θ radians apart, the meter loop GENERATES an E_s field in the meter loop. The result is the meter would not read zero but |ΔV| = ## (θ/2π)\mathcal E ##. I am assuming the meter loop is outside the B flux. To understand why you need to carefully read my Insight blog.

Another example is the battery. The voltmeter reads the emf because, and ONLY because, there is also an E_s field inside the battery. The E_m field does not contribute to the voltmeter reading.

 

[timetraveller123]

ok then thanks once again

 

[Stefan Gustafsson]

That's because voltage by definition is ## V_b - V_a = \int_a^b \vec E_s \cdot d \vec l. ##

Hello, I have found this very interesting thread from the Lewin Videos.

I wonder why you say that voltage by definition is ## V_b - V_a = \int_a^b \vec E_s \cdot d \vec l. ## ?

I would say that as long as you have a conservative total field, the voltage between two points a and b is defined as the scalar potential difference between the two points, which can be calculated as ## V_b - V_a = \int_a^b \vec E \cdot d \vec l. ## Where ## \vec E ## is the total field.

Why would you exclude an E_m field when calculating the voltage?

 

[rude man]

Hello, I have found this very interesting thread from the Lewin Videos.

I wonder why you say that voltage by definition is ## V_b - V_a = \int_a^b \vec E_s \cdot d \vec l. ## ?

I would say that as long as you have a conservative total field, the voltage between two points a and b is defined as the scalar potential difference between the two points, which can be calculated as ## V_b - V_a = \int_a^b \vec E \cdot d \vec l. ## Where ## \vec E ## is the total field.

Why would you exclude an E_m field when calculating the voltage?

Because the definition of voltage is the line integral of the electrostatic field E_s only.
Did you read my Insight article https://www.physicsforums.com/insights/a-new-interpretation-of-dr-walter-lewins-paradox/ ? That might help you understand the difference between electrostatic and electromotive E fields.

 

[Stefan Gustafsson]

Because the definition of voltage is the line integral of the electrostatic field E_s only.
Did you read my Insight article https://www.physicsforums.com/insights/a-new-interpretation-of-dr-walter-lewins-paradox/ ? That might help you understand the difference between electrostatic and electromotive E fields.

Yes, I read your Insight.

Yes, I hear what you are saying about the definition of voltage. I just wonder where you got that definition from?

In the references I have looked in, voltage is simply defined as difference in scalar potential, which is well-defined in any conservative field.
for example: [Field and wave Electromagnetics by David K. Cheng]

 

[rude man]

Yes, I read your Insight.

Yes, I hear what you are saying about the definition of voltage. I just wonder where you got that definition from?

In the references I have looked in, voltage is simply defined as difference in scalar potential, which is well-defined in any conservative field.

Exactly. It is the scalar potential difference. Such a difference exists only in a conservative, i.e. electrostatic field. If an emf field is also present it does not contribute to the scalar potential since it is non-conservative.

I suggest reading the two papers I cited in my Insight article which do a fine job in explaining the situation. Also the Skilling textbook which is I'm sure still available on ebay etc. Skilling is the only one I have encountered so far to emphasize the difference between E_s and E_m and why voltage is the line integral of E_s only.

 

[Stefan Gustafsson]

Exactly. It is the scalar potential difference. Such a difference exists only in a conservative, i.e. electrostatic field. If an emf field is also present it does not contribute to the scalar potential since it is non-conservative.

Well, the total E-field can be conservative in some regions while non-conservative in other regions. For example, think about the Lewin setup, but only look at the right part of the circuit. Imagine a vertical wall between the inductor and the 900Ω resistor.

You are on the right side of the wall. Imagine that you do not even know that the E field comes from a changing magnetic field.

In this region, the total E field is conservative everywhere because there is no changing magnetic field which means that rot E = 0

Since the field is conservative in the whole region, there is a well-defined scalar potential, and you can easily measure voltages using a voltmeter, for example the voltage over the resistor is +0.9V, and the voltage over every part of the conductor is 0V (assuming ideal conductors)

So, what I am trying to say is that as long as the total E field is conservative you always have a well-defined total potential, and thus a well-defined voltage.

I really do not understand why you would want to exclude the "emf field" to get some "true voltage" which is not the same as what you can measure with a voltmeter.

 

[rude man]

1. what is the "Lewin setup"? There are appareently several Lewin lectures and setups existent.
2.

Well, the total E-field can be conservative in some regions while non-conservative in other regions. For example, think about the Lewin setup, but only look at the right part of the circuit. Imagine a vertical wall between the inductor and the 900Ω resistor.

could you reference the "Lewin setup"? I think there are several Lewin setups existent.

Since the field is conservative in the whole region, there is a well-defined scalar potential, and you can easily measure voltages using a voltmeter, for example the voltage over the resistor is +0.9V, and the voltage over every part of the conductor is 0V (assuming ideal conductors).

The voltage in the wires is not zero. It is the net E field that is zero in the wires. There are equal and opposite E fields within the wires: E = E_s - E_m = 0.

Further, it is not easy to read voltages around the circuit with a voltmeter. A voltmeter loop outside the B field will read zero in a wire not because the voltage is zero but because the voltmeter introduces an E_s field on its own. Did you see the Mabilde video? Although his measurement setup only indirectly measures wire voltages, the data is correct. Lewin's statement that "Kirchhoff is wrong" is wrong. The circulation of E_s is always zero. The circulation of E_m = the applied emf. This can be a battery, a Faraday emf, a Seebeck effect device or any other source of emf.

The voltage across a pure inductor is the line integral of the axial E_s field which is why you can read the voltage across an inductor. An equal and opposite E_m field cancels the E_s field so the net E field is zero.

So, what I am trying to say is that as long as the total E field is conservative you always have a well-defined total potential, and thus a well-defined voltage

you are absolutely correct.

 

[rude man]

actually i am coming here from there

why does the voltmeter measure Es not the total E field as you said in the article

Afterthought: I think I have a better answer for you than just "it's by definition":
An emf-generated field has curl: ∇xE_m ≠ 0.
An electrostatic field does not: ∇xE_s = 0.
You will recall that a field with curl cannot have a potential V (a purely mathematical fact: ∇x (∇V) = 0).
Therefore, an E_m field cannot have a potential, i.e. a voltage V.

 

[timetraveller123]

no i know that
it is possible to know which is Em and which is Es when you look at one entire loop
but if you look at a open curve like,voltmeter connected between two points, how is it supposed to differentiate the two fields
or does this have to do with the inner workings of the voltmeter ?

 

[Stefan Gustafsson]

no i know that
it is possible to know which is Em and which is Es when you look at one entire loop
but if you look at a open curve like,voltmeter connected between two points, how is it supposed to differentiate the two fields
or does this have to do with the inner workings of the voltmeter ?

I disagree with @rude man .

A voltmeter really measures the current that passes through the voltmeter. Since the internal resistance of the voltmeter is known (very high) a voltage can be calculated using Ohms law, V = R*I

The voltmeter does not know anything about a separation between E_m and E_s.

To calculate the voltage displayed by a voltmeter you can always use Faraday's law.

Normally the voltmeter will display the voltage between the ends of the measuring probes, but if there is a changing magnetic flux in the loop formed by the measuring wires and the measured object there is also an induced emf that you need to account for.

 

[rude man]

no i know that
it is possible to know which is Em and which is Es when you look at one entire loop
but if you look at a open curve like,voltmeter connected between two points, how is it supposed to differentiate the two fields
or does this have to do with the inner workings of the voltmeter ?

∇xE doesn't apply just to a closed loop. The curl exists (or doesn't exist) at every point. Forget closed loops for a minute. The potential difference between any two points is the line integral of E_s between the two points. The curve connecting the two points is immaterial. That's why it's called a 'conservative' field. We would like the voltmeter to read the potential difference between any two points.

In the case of the battery-resistor circuit there is no problem: the voltmeter will read the potential difference between any two points along the circuit including across the battery because remember there are two E fields inside the battery and the voltmeter reads line-integrated E_s only. Check my paper if you need to.

When there is magnetic induction there's a problem because the voltmeter circuit forms a section of an alternate closed loop. Take a one-turn coil of resistance R and radius a. You get pure E_m field in the coil by Maxwell's ∇xE_m = -∂B/∂t. No E_s, no potential drop anywhere. Accordingly, we might expect a voltmeter, if connected, to read zero. But when the voltmeter is connected between two points along the coil the voltmeter, forming an alternate magnetic closed loop, will read a finite voltage. If the voltmeter circuit is outside the B field that voltage will be emf*s/2πa with s the length of the coil between the two points.

So bottom line, in the magnetic induction case there is no potential difference anywhere but the voltmeter will nonetheless read a potential difference since it is included in the magnetic loop.

PS you will have to decide between myself and @Stefan Gustafsson.